CH ALLENGE Calculating sample sizes for the twosample problem. For a single proportion the margin of error of a confidence interval is largest for any given sample size n and confidence level C when ˆp = 0.5. This led us to use p∗ = 0.5 for planning purposes. The same kind of result is true for the two-sample problem. The margin of error of the confidence interval for the difference between two proportions is largest when ˆp1 = ˆp2 = 0.5.

You are planning a survey and will calculate a 95% confidence interval for the difference in two proportions when the data are collected. You would like the margin of error of the interval to be less than or equal to 0.075. You will use the same sample size n for both populations.

(a) How large a value of n is needed?

(b) Give a general formula for n in terms of the desired margin of error m and the critical value z∗.